Question

In the given figure, AB and CD are two equal chords of a circle with centre O. OP and OQ are perpendicular on chords AB and CD, respectively. If ∠POQ = 150^∘, then ∠APQ = __________.

Answer 1

Given:
AB = CD
OP ⊥ AB and OQ ⊥ CD
∠POQ = 150° ...(1)


In ∆POQ,
OP = OQ (equal chords are equidistant from the centre)
∴ ∠OPQ = ∠OQP (angles opposite to equal sides are equal) ...(2)

Now, ∠OPQ + ∠OQP + ∠POQ = 180° (angle sum property)
⇒ 2∠OPQ + 150° = 180° (From (1) and (2))
⇒ 2∠OPQ = 180° − 150°
⇒ 2∠OPQ = 30°
⇒ ∠OPQ = 15° ...(3)


Since, OP ⊥ AB
Thus, ∠OPA = 90° ....(4)


Now, ∠OPA = ∠OPQ + ∠APQ
⇒ 90° = 15° + ∠APQ (From (3) and (4))
⇒ ∠APQ = 90° − 15°
⇒ ∠APQ = 75°


Hence, ∠APQ = 75°.